The `law of small numbers' in the title, endearingly revived from that of the (in)famous 1898 monograph Das Gesetz der kleinen Zahlen by Ladislaus von Bortkiewicz, refers to the classical approximation of the Binomial(n,p) distribution by the Poisson distribution ...
Three leading figures of extreme value and point process theory joined forces to deliver a week-long Deutsche Mathematiker Vereinigung Seminar under the same title in 1991. The aim of the series is to give a coherent introduction to a given research field, under current vigorous development, to young researchers, who are typically assistants at German universities either before their PhD or immediately after. The resulting truly excellent book is their extended and polished lecture notes, completely unified. Many of the proofs are not given, but the nature and the essence of the results, along with their motivations and roles in applied and theoretical problems are masterfully sketched throughout and complete references are given for the details. The lovely scenario above is a condensed form of the author's heuristics from their extremely nice description in the first chapter.
Part I of the book (The iid case: functional laws of samll numbers, pp. 1-153) is what concentrates on the program given above more closely. Chapter 1 (Laws of small numbers, pp. 1-24) makes this program precise, emphasizing the basic function of the Hellinger distance, through which the approximations are done, since in the iid setting it gives better bounds in terms of the underlying distribution than the variational distance. Chapter 2 (Extreme value theory, pp. 25-69) and Chap. 3 (Estimation of conditional curves, pp. 71-104) executes the program delineated concerning applications to the univariate peaks over threshold method and for the estimation of regression and related conditional functions in various nonparametric and semiparametric situations, respectively. The former describes many important recent and new applications, mainly due to the authors, for estimating parameters of generalized Pareto and other extreme value distributions or those in their domains of attraction. Particularly interesting to this reviewer are the results on estimating the class index (one of three) of an attracting extreme value distribution. Chapters 4 and 5 (Multivariate maxima and Mulivariate extremes, pp. 105-135) provide the elements of basic multivariate extreme value theory and the corresponding extension of the peaks over threshold method, respectively, the latter being a recent result of one of the authors. It is important that the book comes with a disk containing the statistical software package XTREMES, developed by S. Hassmann, R.-D. Reiss and M. Thomas and which runs on IBM-compatible personal computers under MS-DOS or compatible operating systems. While Chap. 6 (Extreme value analysis with XTREMES, pp. 137-153) gives examples of simulation and exploratory data analysis using the package, an Appendix (pp.255-274) is a detailed User's Guide to XTREMES written by the three developers.
Part II (Non iid observations, pp. 155-254) opens with another attractive introduction (Chap. 7, pp. 157-167) to the non-iid case. Chapter 8 (Extremes of random sequences, pp. 169-189) presents a very general exceedance theory and applies it to stationary (and some examples of nonstationary) sequences as well as to independent but not identically distributed sequences. Chapter 9 (Extremes of Gaussian processes, pp. 191-204) deals with both the stationary and the nonstationary case, and a nice application to empirical characteristic functions. Chapter 10 (Extensions to rare events, pp. 205-235) sketches a general unified theory of triangular arrays of rare events and point process exceedances that is capable of handling multivariate nonstationary sequences as an applicaton. Finally, Chap. 11 (Statistics of extremes, pp. 237-254) contains the analyses of some concrete nonstationary time series data sets from ecology and agriculture.
No doubt, this field is really under very vigorous current development, and the present book will greatly contribute to this process and its success. It is XTREMELY lucidly written, and hence it is probably the best introduction currently available to doing research here. It is also very carefully written and produced; practically error-free. (A misprint is having I. Csiszár incorrectly as L. Csizár, easy to detect for the Hungarian eye.) Does anyone know why von Bortkiewicz changed the spelling of his name, sometime between 1895 and 1898, from Bortkewitsch?